NCERT Class 10 Maths Solutions: Probability

Question:

Are the arguments in the following sentence correct or not correct? Give a reason for your answer.

If two coins are tossed simultaneously there are three possible outcomes, two heads, two tails or one of each. Therefore, for each of these outcomes, the probability is $13.$

A. Correct

B. Incorrect

Concept in a Minute:

The question deals with probability and sample space. The sample space is the set of all possible outcomes of an experiment. Probability of an event is calculated as the number of favorable outcomes divided by the total number of possible outcomes. It’s crucial to correctly identify all possible outcomes in the sample space, especially when multiple events are combined.

Explanation:

The statement is incorrect. When two coins are tossed simultaneously, the possible outcomes are:
1. Head, Head (HH)
2. Head, Tail (HT)
3. Tail, Head (TH)
4. Tail, Tail (TT)

Therefore, there are four possible outcomes, not three.

Let’s analyze the outcomes mentioned in the question:
– Two heads: This corresponds to HH. There is 1 such outcome.
– Two tails: This corresponds to TT. There is 1 such outcome.
– One of each: This corresponds to HT and TH. There are 2 such outcomes.

So, the statement incorrectly groups HT and TH into a single outcome “one of each” and also omits the distinct possibility of HT and TH leading to a total of only three listed outcomes.

The probability of each outcome in the correct sample space of four outcomes is:
– Probability of HH = 1/4
– Probability of HT = 1/4
– Probability of TH = 1/4
– Probability of TT = 1/4

If we consider the event “two heads”, its probability is 1/4.
If we consider the event “two tails”, its probability is 1/4.
If we consider the event “one of each” (meaning either HT or TH), its probability is 2/4 = 1/2.

The statement incorrectly claims there are three possible outcomes and then assigns a probability of 1/3 to each. This is wrong because the sample space is larger (four outcomes) and the probabilities are not equal for all the outcomes as described in the question’s simplified outcome grouping.

The final answer is $\boxed{B}$.

Question:

Which of the following arguments are correct and which are not correct? Give a reason for your answer.

If a die is thrown, there are two possible outcomes, an odd number or an even number. Therefore, the probability of getting an odd number is 1/2.

A. Correct

B. Incorrect

Concept in a Minute:

Probability is defined as the ratio of favorable outcomes to the total number of possible outcomes. For a fair die, each face has an equal chance of appearing.

Explanation:

The statement is incorrect. While it is true that when a die is thrown, the outcomes can be classified as odd or even, the assumption that these are the only two possible outcomes in terms of their probability is flawed. A standard die has six faces numbered 1, 2, 3, 4, 5, and 6. The odd numbers are 1, 3, and 5, which are 3 favorable outcomes. The even numbers are 2, 4, and 6, which are also 3 favorable outcomes. Therefore, the total number of possible outcomes is 6. The probability of getting an odd number is the number of odd outcomes divided by the total number of outcomes, which is 3/6 = 1/2. The statement is correct that the probability of getting an odd number is 1/2, but the reasoning given is incomplete and potentially misleading if not fully understood. The argument that “there are two possible outcomes, an odd number or an even number” implies that these are mutually exclusive and exhaustive categories, which is true. However, the “therefore” connects these categories directly to a probability of 1/2 without explicitly stating that the number of odd and even outcomes are equal. If the question implies that the *only* two possibilities are odd and even, and that these are the only things to consider for probability without counting, then it would be incorrect. But if it means that the set of outcomes can be partitioned into odd and even, and then it calculates the probability correctly based on the number of elements in each partition, then it is correct. Given the phrasing, it is generally accepted as correct in the context of basic probability.

The correct answer is A. Correct.

Question:

A piggy bank contains hundred 50 p coins, fifty Rs 1 coins, twenty Rs 2 coins and ten Rs 5 coins. If it is equally likely that one of the coins will fall out when the bank is turned upside down, what is the probability that the coin will not be a Rs. 5 coin?

Concept in a Minute:

Probability is the ratio of the number of favorable outcomes to the total number of possible outcomes. Favorable outcomes are the events that satisfy the condition we are interested in. Total possible outcomes are all the possible events that can occur.

Explanation:

First, we need to find the total number of coins in the piggy bank.
Number of 50 p coins = 100
Number of Rs 1 coins = 50
Number of Rs 2 coins = 20
Number of Rs 5 coins = 10
Total number of coins = 100 + 50 + 20 + 10 = 180

Next, we need to find the number of coins that are NOT Rs 5 coins.
Number of coins that are not Rs 5 coins = (Number of 50 p coins) + (Number of Rs 1 coins) + (Number of Rs 2 coins)
Number of coins that are not Rs 5 coins = 100 + 50 + 20 = 170

The probability of an event is calculated as:
Probability = (Number of favorable outcomes) / (Total number of possible outcomes)

In this case, the favorable outcome is picking a coin that is not a Rs 5 coin.
The total possible outcome is picking any coin from the piggy bank.

Probability (coin will not be a Rs 5 coin) = (Number of coins that are not Rs 5 coins) / (Total number of coins)
Probability (coin will not be a Rs 5 coin) = 170 / 180

Now, we simplify the fraction:
170 / 180 = 17 / 18

Alternatively, we can calculate the probability of picking a Rs 5 coin first and then subtract it from 1.
Number of Rs 5 coins = 10
Total number of coins = 180
Probability (coin will be a Rs 5 coin) = 10 / 180 = 1 / 18

Probability (coin will not be a Rs 5 coin) = 1 – Probability (coin will be a Rs 5 coin)
Probability (coin will not be a Rs 5 coin) = 1 – (1 / 18)
Probability (coin will not be a Rs 5 coin) = (18 / 18) – (1 / 18) = 17 / 18

The probability that the coin will not be a Rs 5 coin is 17/18.

Question:

Following experiment have equally likely outcomes? Explain.

A driver attempts to start a car. The car starts or does not start.

Concept in a Minute:

Equally Likely Outcomes: An experiment has equally likely outcomes if each possible outcome has the same probability of occurring. In simpler terms, no outcome is more likely to happen than any other.
Probability: The likelihood of a specific event happening, calculated as the number of favorable outcomes divided by the total number of possible outcomes.

Explanation:

The question asks whether the outcomes of a car starting or not starting when a driver attempts to start it are equally likely.

To determine if the outcomes are equally likely, we need to consider if the car starting and the car not starting have the same probability of occurring.

In reality, whether a car starts or not depends on many factors such as the car’s condition (battery charge, fuel level, engine health), the weather, and mechanical issues. These factors do not inherently make the two outcomes (starting or not starting) have an equal chance of happening.

For example, a car that is well-maintained and has a good battery is much more likely to start than a car that is old, has a dead battery, or is out of fuel. Conversely, if a car has a known engine problem, it might be more likely *not* to start.

Therefore, the outcomes “the car starts” and “the car does not start” are not necessarily equally likely. Their probabilities depend on the specific condition of the car at the time of the attempt.

Since the outcomes are not guaranteed to have the same probability, they are not equally likely.

Final Answer: No, the experiment does not have equally likely outcomes. The probability of a car starting or not starting depends on various external factors and the condition of the car, not on an inherent equal chance for each outcome.

Question:

A box contains 5 red marbles, 8 white marbles and 4 green marbles. One marble is taken out of the box at randam.What is the probability that the marble taken out wil be not green?

Concept in a Minute:

Probability is defined as the ratio of the number of favorable outcomes to the total number of possible outcomes.
Total number of outcomes = Sum of all individual outcomes.
Number of favorable outcomes = Number of outcomes that satisfy the given condition.

Explanation:

The question asks for the probability that the marble taken out will be “not green”. This means the marble can be either red or white.

First, let’s find the total number of marbles in the box.
Number of red marbles = 5
Number of white marbles = 8
Number of green marbles = 4
Total number of marbles = Number of red marbles + Number of white marbles + Number of green marbles
Total number of marbles = 5 + 8 + 4 = 17

Next, let’s find the number of marbles that are “not green”. These are the red and white marbles.
Number of marbles that are not green = Number of red marbles + Number of white marbles
Number of marbles that are not green = 5 + 8 = 13

Now, we can calculate the probability of taking out a marble that is not green.
Probability (Not Green) = (Number of marbles that are not green) / (Total number of marbles)
Probability (Not Green) = 13 / 17

Alternatively, we can calculate the probability of taking out a green marble and subtract it from 1.
Probability (Green) = (Number of green marbles) / (Total number of marbles)
Probability (Green) = 4 / 17

Probability (Not Green) = 1 – Probability (Green)
Probability (Not Green) = 1 – (4 / 17)
Probability (Not Green) = (17 / 17) – (4 / 17)
Probability (Not Green) = (17 – 4) / 17
Probability (Not Green) = 13 / 17

The probability that the marble taken out will be not green is 13/17.

Question:

A box contains 90 discs which are numbered from 1 to 90. If one disc is drawn at random from the box, find the probability that it bears a two-digit number.

Concept in a Minute:

Explanation:

The problem asks for the probability of drawing a disc with a two-digit number from a box containing discs numbered from 1 to 90.

Step 1: Determine the total number of possible outcomes.
The discs are numbered from 1 to 90, so there are 90 possible outcomes when one disc is drawn at random.

Step 2: Determine the number of favorable outcomes.
A favorable outcome is drawing a disc with a two-digit number.
The one-digit numbers are from 1 to 9.
The two-digit numbers start from 10 and go up to 90.
To find the count of two-digit numbers from 10 to 90, we can subtract the count of one-digit numbers from the total number of discs.
Number of one-digit numbers = 9 (from 1 to 9)
Number of two-digit numbers = Total number of discs – Number of one-digit numbers
Number of two-digit numbers = 90 – 9 = 81.
Alternatively, we can count them directly: from 10 to 90 inclusive. The number of integers in a range [a, b] is b – a + 1. So, 90 – 10 + 1 = 81.

Step 3: Calculate the probability.
Probability of drawing a two-digit number = (Number of two-digit numbers) / (Total number of discs)
Probability = 81 / 90

Step 4: Simplify the probability (if possible).
Both 81 and 90 are divisible by 9.
81 ÷ 9 = 9
90 ÷ 9 = 10
So, the simplified probability is 9/10.

The probability that the drawn disc bears a two-digit number is 9/10.

Question:

A child has a die whose six faces shows the letters as given below:

The die is thrown once. What is the probability of getting (i) A? (ii) D?

Concept in a Minute:

Probability is the measure of the likelihood of an event occurring. It is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes.
P(Event) = (Number of favorable outcomes) / (Total number of possible outcomes)

Explanation:

The die has six faces, and each face shows a letter. We need to determine the probability of getting a specific letter when the die is thrown once.

First, let’s identify the total number of possible outcomes. Since the die has six faces, there are 6 possible outcomes when it is thrown once.

(i) Probability of getting A:
To find the probability of getting ‘A’, we need to count how many faces show the letter ‘A’.
Looking at the given letters on the faces: A, B, C, D, E, A.
There are 2 faces that show the letter ‘A’.
So, the number of favorable outcomes for getting ‘A’ is 2.
The probability of getting ‘A’ is:
P(A) = (Number of faces with A) / (Total number of faces) = 2 / 6
This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 2.
P(A) = 1 / 3

(ii) Probability of getting D:
To find the probability of getting ‘D’, we need to count how many faces show the letter ‘D’.
Looking at the given letters on the faces: A, B, C, D, E, A.
There is 1 face that shows the letter ‘D’.
So, the number of favorable outcomes for getting ‘D’ is 1.
The probability of getting ‘D’ is:
P(D) = (Number of faces with D) / (Total number of faces) = 1 / 6

Therefore, the probability of getting (i) A is 1/3, and the probability of getting (ii) D is 1/6.

Question:

A game consists of tossing a one rupee coin 3 times and noting its outcome each time. Hanif wins if all the tosses give the same result, i.e., three heads or three tails, and loses otherwise. Calculate the probability that Hanif will lose the game.

Concept in a Minute:

Probability is the measure of the likelihood of an event occurring. It is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes. For multiple independent events, the total number of outcomes is the product of the number of outcomes for each event.

Explanation:

Step 1: Identify all possible outcomes when a coin is tossed 3 times.
Each toss has two possible outcomes: Heads (H) or Tails (T). Since the coin is tossed 3 times, the total number of possible outcomes is 2 * 2 * 2 = 8.
The possible outcomes are: HHH, HHT, HTH, THH, HTT, THT, TTH, TTT.

Step 2: Identify the outcomes where Hanif wins.
Hanif wins if all the tosses give the same result, meaning three heads (HHH) or three tails (TTT).
So, there are 2 winning outcomes.

Step 3: Calculate the probability of Hanif winning.
Probability of winning = (Number of winning outcomes) / (Total number of possible outcomes)
Probability of winning = 2 / 8 = 1/4.

Step 4: Calculate the probability of Hanif losing.
Hanif loses if he does not win. The event of losing is the complement of the event of winning.
Probability of losing = 1 – (Probability of winning)
Probability of losing = 1 – 1/4 = 3/4.

Alternatively, we can count the losing outcomes directly from Step 1.
The losing outcomes are: HHT, HTH, THH, HTT, THT, TTH.
There are 6 losing outcomes.
Probability of losing = (Number of losing outcomes) / (Total number of possible outcomes)
Probability of losing = 6 / 8 = 3/4.

Final Answer: The probability that Hanif will lose the game is 3/4.

Question:

A bag contains 3 red balls and 5 black balls. A ball is drawn at random from the bag. What is the probability that the ball drawn is red?

Concept in a Minute:

Probability is the measure of the likelihood that an event will occur. It is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes. The formula for probability is:
P(Event) = (Number of favorable outcomes) / (Total number of possible outcomes)

Explanation:

The question asks for the probability of drawing a red ball from a bag containing red and black balls.
First, identify the number of favorable outcomes, which is the number of red balls.
Second, identify the total number of possible outcomes, which is the sum of red balls and black balls.
Then, apply the probability formula.

Step-by-step derivation:
1. Identify the number of red balls in the bag.
Number of red balls = 3

2. Identify the number of black balls in the bag.
Number of black balls = 5

3. Calculate the total number of balls in the bag.
Total number of balls = Number of red balls + Number of black balls
Total number of balls = 3 + 5 = 8

4. Determine the number of favorable outcomes, which is drawing a red ball.
Number of favorable outcomes = Number of red balls = 3

5. Determine the total number of possible outcomes, which is drawing any ball from the bag.
Total number of possible outcomes = Total number of balls = 8

6. Calculate the probability of drawing a red ball using the probability formula:
P(Red ball) = (Number of favorable outcomes) / (Total number of possible outcomes)
P(Red ball) = 3 / 8

Final Answer: The probability that the ball drawn is red is 3/8.

Question:

A bag contains 3 red balls and 5 black balls. A ball is drawn at random from the bag. What is the probability that the ball drawn is not red?

Concept in a Minute:

Probability is the measure of the likelihood of an event occurring. It is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes.
Probability (Event) = (Number of favorable outcomes) / (Total number of possible outcomes)
The complement of an event is an event that does not occur. The probability of the complement of an event A, denoted as P(A’), is given by P(A’) = 1 – P(A).

Explanation:

First, identify the total number of balls in the bag.
Total number of red balls = 3
Total number of black balls = 5
Total number of balls = Number of red balls + Number of black balls = 3 + 5 = 8

Next, determine the number of outcomes that are *not* red. In this case, this means the number of black balls.
Number of balls that are not red (i.e., black balls) = 5

Now, calculate the probability of drawing a ball that is not red.
Probability (not red) = (Number of balls that are not red) / (Total number of balls)
Probability (not red) = 5 / 8

Alternatively, you can first calculate the probability of drawing a red ball.
Probability (red) = (Number of red balls) / (Total number of balls)
Probability (red) = 3 / 8

Then, use the concept of complementary probability. The event “not red” is the complement of the event “red”.
Probability (not red) = 1 – Probability (red)
Probability (not red) = 1 – (3 / 8)
Probability (not red) = (8 / 8) – (3 / 8)
Probability (not red) = 5 / 8

The probability that the ball drawn is not red is 5/8.

Question:

A lot consists of 144 ball pens of which 20 are defective and the others are good. Nuri will buy a pen if it is good, but will not buy if it is defective. The shopkeeper draws one pen at random and gives it to her. What is the probability that she will buy it?

Concept in a Minute:

Probability: The ratio of the number of favorable outcomes to the total number of possible outcomes.

Explanation:

The question asks for the probability that Nuri will buy a pen. Nuri will buy a pen only if it is good. Therefore, the favorable outcome is selecting a good pen.

First, identify the total number of pens in the lot.
Total number of pens = 144

Next, identify the number of defective pens.
Number of defective pens = 20

Now, calculate the number of good pens.
Number of good pens = Total number of pens – Number of defective pens
Number of good pens = 144 – 20
Number of good pens = 124

The probability of an event is calculated as:
Probability (Event) = (Number of favorable outcomes) / (Total number of possible outcomes)

In this case, the event is Nuri buying a pen, which means selecting a good pen.
Number of favorable outcomes (Nuri buying a pen) = Number of good pens = 124
Total number of possible outcomes (Total pens to choose from) = 144

So, the probability that Nuri will buy a pen is:
Probability (Nuri buys a pen) = (Number of good pens) / (Total number of pens)
Probability (Nuri buys a pen) = 124 / 144

This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor. Both 124 and 144 are divisible by 4.
124 ÷ 4 = 31
144 ÷ 4 = 36

Therefore, the simplified probability is 31/36.

The final answer is $\boxed{31/36}$.

Question:

A bag contains lemon flavoured candies only. Malini takes out one candy without looking into the bag. What is the probability that she takes out a lemon flavoured candy?

Concept in a Minute:

Probability is the measure of the likelihood that an event will occur. It is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes.

Explanation:

The question states that the bag contains lemon flavoured candies only. This means that every single candy in the bag is lemon flavoured. When Malini takes out one candy without looking, there are no other types of candies she could possibly pick. Therefore, the event of picking a lemon flavoured candy is certain to happen.

In terms of probability, the number of favorable outcomes (picking a lemon flavoured candy) is equal to the total number of possible outcomes (the total number of candies in the bag).

Let F be the event of picking a lemon flavoured candy.
Let N(F) be the number of favorable outcomes (number of lemon flavoured candies).
Let N(S) be the total number of possible outcomes (total number of candies in the bag).

According to the problem, the bag contains lemon flavoured candies only.
So, the number of lemon flavoured candies = Total number of candies in the bag.
N(F) = N(S)

The probability of an event is given by:
P(F) = N(F) / N(S)

Since N(F) = N(S), we can substitute this into the probability formula:
P(F) = N(S) / N(S)
P(F) = 1

A probability of 1 indicates that the event is certain to occur.

Therefore, the probability that Malini takes out a lemon flavoured candy is 1.

Question:

A box contains 90 discs which are numbered from 1 to 90. If one disc is drawn at random from the box, find the probability that it bears a number divisible by 5.

Concept in a Minute:

Probability is defined as the ratio of the number of favorable outcomes to the total number of possible outcomes.
Favorable outcome: An event that is desired or likely to happen.
Total outcomes: All possible results of an experiment.

Explanation:

The question asks for the probability of drawing a disc with a number divisible by 5 from a box containing discs numbered from 1 to 90.

Step 1: Identify the total number of possible outcomes.
The discs are numbered from 1 to 90, so there are 90 discs in the box.
Therefore, the total number of possible outcomes when drawing one disc is 90.

Step 2: Identify the number of favorable outcomes.
A favorable outcome is drawing a disc with a number divisible by 5.
We need to find how many numbers between 1 and 90 are divisible by 5.
These numbers are 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90.
To find the count directly, we can divide the largest number by 5: 90 / 5 = 18.
So, there are 18 numbers divisible by 5 between 1 and 90.
The number of favorable outcomes is 18.

Step 3: Calculate the probability.
Probability = (Number of favorable outcomes) / (Total number of possible outcomes)
Probability = 18 / 90

Step 4: Simplify the probability.
Both 18 and 90 are divisible by 18.
18 / 18 = 1
90 / 18 = 5
So, the simplified probability is 1/5.

The probability that the disc bears a number divisible by 5 is 1/5.

Question:

A box contains 90 discs which are numbered from 1 to 90. If one disc is drawn at random from the box, find the probability that it bears a perfect square number.

Concept in a Minute:

Probability of an event = (Number of favorable outcomes) / (Total number of possible outcomes)
Identifying perfect squares within a given range.

Explanation:

The problem asks for the probability of drawing a disc with a perfect square number from a box containing discs numbered from 1 to 90.

First, we need to determine the total number of possible outcomes. Since the discs are numbered from 1 to 90, there are 90 possible outcomes when one disc is drawn at random.

Next, we need to identify the number of favorable outcomes, which are the discs bearing perfect square numbers between 1 and 90. Let’s list the perfect squares:
1^2 = 1
2^2 = 4
3^2 = 9
4^2 = 16
5^2 = 25
6^2 = 36
7^2 = 49
8^2 = 64
9^2 = 81
10^2 = 100 (This is greater than 90, so it is not included).

So, the perfect square numbers between 1 and 90 are 1, 4, 9, 16, 25, 36, 49, 64, and 81.
There are 9 perfect square numbers. Therefore, the number of favorable outcomes is 9.

Now, we can calculate the probability using the formula:
Probability (Perfect Square) = (Number of perfect square numbers) / (Total number of discs)
Probability (Perfect Square) = 9 / 90

This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 9.
Probability (Perfect Square) = 1 / 10

Thus, the probability that the disc drawn bears a perfect square number is 1/10.

Question:

One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting a red face card.

Concept in a Minute:

Understanding probability as the ratio of favorable outcomes to the total possible outcomes. Identifying specific card types within a standard deck.

Explanation:

A standard deck of 52 cards has 4 suits: hearts, diamonds, clubs, and spades. Hearts and diamonds are red suits, while clubs and spades are black suits. Each suit has 13 cards: Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, and King. Face cards are Jack, Queen, and King.

We are looking for red face cards.
The red suits are hearts and diamonds.
The face cards in hearts are Jack of Hearts, Queen of Hearts, and King of Hearts. (3 cards)
The face cards in diamonds are Jack of Diamonds, Queen of Diamonds, and King of Diamonds. (3 cards)
So, there are a total of 3 + 3 = 6 red face cards.

The total number of cards in the deck is 52.

The probability of an event is calculated as:
Probability = (Number of favorable outcomes) / (Total number of possible outcomes)

In this case:
Number of favorable outcomes (red face cards) = 6
Total number of possible outcomes (total cards) = 52

Probability of getting a red face card = 6 / 52

This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 2.
6 ÷ 2 = 3
52 ÷ 2 = 26

So, the probability of getting a red face card is 3/26.

The final answer is $\boxed{3/26}$.

Question:

The probability of an event that is certain to happen is ______. Such as event is called ______.

Concept in a Minute:

Probability: Probability is a measure of the likelihood of an event occurring. It is expressed as a number between 0 and 1, inclusive.
Certain Event: An event that is guaranteed to happen is called a certain event.
Impossible Event: An event that cannot happen is called an impossible event.

Explanation:

The probability of an event that is certain to happen is 1. This is because a certain event will always occur, and its probability is the highest possible value. Such an event is called a certain event.

Detailed Solution Structure:
1. Understand the definition of probability: Recall that probability quantifies the chance of an event occurring, with values ranging from 0 (impossible) to 1 (certain).
2. Consider an event that is certain to happen: Think about what “certain to happen” means in terms of its likelihood.
3. Assign a numerical value to this likelihood: Based on the definition of probability, what numerical value represents something that is guaranteed to occur?
4. Identify the term used for such an event: What is the specific name given to an event that is certain to happen?
5. Fill in the blanks: Use the answers from steps 3 and 4 to complete the given sentence.

Answer:
The probability of an event that is certain to happen is 1. Such as event is called a certain event.

Question:

The sum of the probabilities of all the elementary events of an experiment is _________.

Concept in a Minute:

Probability of an event is a measure of the likelihood of that event occurring. For any experiment, the set of all possible outcomes is called the sample space. Elementary events are the individual, most basic outcomes within the sample space. The axioms of probability state that the probability of any event is between 0 and 1 (inclusive), and the sum of the probabilities of all mutually exclusive and exhaustive events (which collectively form the sample space) is equal to 1.

Explanation:

An experiment can be broken down into its elementary events, which are the simplest possible outcomes. For example, in the experiment of tossing a coin once, the elementary events are ‘getting a head’ and ‘getting a tail’. In the experiment of rolling a die once, the elementary events are ‘getting a 1’, ‘getting a 2’, ‘getting a 3’, ‘getting a 4’, ‘getting a 5’, and ‘getting a 6’.

The sample space of an experiment is the set of all possible elementary events. According to the fundamental axioms of probability, the sum of the probabilities of all these mutually exclusive and exhaustive elementary events must be equal to 1. This is because one of these elementary events is guaranteed to occur when the experiment is performed.

Therefore, the sum of the probabilities of all the elementary events of an experiment is 1.

Question:

A lot of 20 bulbs contain 4 defective ones. One bulb is drawn at random from the lot. What is the probability that this bulb is defective?
Suppose the bulb drawn in (1) is not defective and is not replaced. Now one bulb is drawn at random from the rest. What is the probability that this bulb is not defective?

Concept in a Minute:

Probability is the measure of the likelihood of an event occurring. It is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes. Conditional probability is the probability of an event occurring given that another event has already occurred.

Explanation:

Part 1:
Total number of bulbs in the lot = 20
Number of defective bulbs = 4
Number of non-defective bulbs = 20 – 4 = 16

The probability of drawing a defective bulb is the number of defective bulbs divided by the total number of bulbs.
Probability (defective bulb) = (Number of defective bulbs) / (Total number of bulbs)
Probability (defective bulb) = 4 / 20 = 1/5

Part 2:
Given that the bulb drawn in the first draw was not defective and it was not replaced.
So, after the first draw, the total number of bulbs remaining in the lot is 20 – 1 = 19.
Since the bulb drawn was not defective, the number of defective bulbs remains the same.
Number of defective bulbs remaining = 4
Number of non-defective bulbs remaining = 16 – 1 = 15

Now, we need to find the probability that the bulb drawn at random from the rest is not defective.
Probability (not defective bulb from the rest) = (Number of non-defective bulbs remaining) / (Total number of bulbs remaining)
Probability (not defective bulb from the rest) = 15 / 19

Question:

A lot consists of 144 ball pens, of which 20 are defective and the others are good. Nuri will buy a pen if it is good, but will not buy if it is defective. The shopkeeper draws one pen at random and gives it to her. What is the probability that she will not buy it?

Concept in a Minute:

Explanation:

Here, the event we are interested in is that Nuri will not buy the pen. Nuri will not buy a pen if it is defective.
Total number of ball pens in the lot = 144
Number of defective pens = 20
Number of good pens = 144 – 20 = 124

The shopkeeper draws one pen at random. The total number of possible outcomes is the total number of pens, which is 144.

The favorable outcome for Nuri not buying the pen is drawing a defective pen.
Number of favorable outcomes (defective pens) = 20

The probability that Nuri will not buy the pen is the probability of drawing a defective pen.
P(Nuri will not buy) = P(drawing a defective pen) = (Number of defective pens) / (Total number of pens)
P(Nuri will not buy) = 20 / 144

To simplify the fraction:
Divide both numerator and denominator by their greatest common divisor.
20 = 2 * 2 * 5
144 = 2 * 2 * 2 * 2 * 3 * 3
The greatest common divisor is 2 * 2 = 4.

P(Nuri will not buy) = 20 ÷ 4 / 144 ÷ 4 = 5 / 36

Therefore, the probability that she will not buy it is 5/36.

Question:

Following experiment have equally likely outcomes? Explain.

A player attempts to shoot a basketball. She/he shoots or misses the shot.

Concept in a Minute:

Equally likely outcomes are those in which each outcome has the same probability of occurring. To determine if outcomes are equally likely, we need to consider the nature of the event and if there’s any bias or inherent advantage for one outcome over another.

Explanation:

In the given experiment, a player attempts to shoot a basketball. The two possible outcomes are that the player either shoots (makes the shot) or misses the shot.

To determine if these outcomes are equally likely, we need to assess the probability of each. The probability of a basketball player making a shot depends on several factors, including:
1. The player’s skill level.
2. The difficulty of the shot.
3. The pressure of the situation.
4. Whether the player is having a good or bad day.

There is no inherent reason why a player is equally likely to make or miss a shot in every instance. For a highly skilled player, the probability of making a shot might be significantly higher than missing. Conversely, a novice player might have a higher probability of missing.

Since the probability of shooting and missing the shot is not necessarily equal for any given player, these outcomes are not equally likely in general.

Therefore, the experiment does not have equally likely outcomes.

Question:

One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting the jack of hearts.

Concept in a Minute:

Explanation:

A standard deck of cards has 52 cards. These cards are divided into four suits: hearts, diamonds, clubs, and spades. Each suit contains 13 cards: Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, and King.

The question asks for the probability of drawing the “jack of hearts”.
First, let’s identify the total number of possible outcomes. When drawing one card from a well-shuffled deck of 52 cards, there are 52 possible outcomes.
Total number of possible outcomes = 52.

Next, let’s identify the number of favorable outcomes. The favorable outcome is drawing the specific card, which is the jack of hearts. In a standard deck of 52 cards, there is only one jack of hearts.
Number of favorable outcomes = 1.

Now, we can calculate the probability using the formula:
Probability of getting the jack of hearts = (Number of favorable outcomes) / (Total number of possible outcomes)
Probability of getting the jack of hearts = 1 / 52.

Therefore, the probability of getting the jack of hearts is 1/52.

Question:

One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting a face card.

Concept in a Minute:

Probability is calculated as the ratio of favorable outcomes to the total possible outcomes. In a deck of cards, understanding the different types of cards (face cards, numbered cards, suits) is crucial for determining favorable outcomes.

Explanation:

A standard deck of 52 cards contains 4 suits: hearts, diamonds, clubs, and spades. Each suit has 13 cards: Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, and King.
Face cards are the Jack, Queen, and King.
In each suit, there are 3 face cards.
Since there are 4 suits, the total number of face cards in a deck is 3 face cards/suit * 4 suits = 12 face cards.
The total number of possible outcomes when drawing one card from a well-shuffled deck is 52 (the total number of cards).
The number of favorable outcomes (getting a face card) is 12.
The probability of an event is given by the formula:
Probability (Event) = (Number of favorable outcomes) / (Total number of possible outcomes)
Therefore, the probability of getting a face card is:
P(Face Card) = 12 / 52
This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 4.
12 ÷ 4 = 3
52 ÷ 4 = 13
So, the simplified probability is 3/13.

Question:

12 defective pens are accidentally mixed with 132 good ones. It is not possible to just look at a pen and tell whether or not it is defective. One pen is taken out at random from this lot. Determine the probability that the pen taken out is a good one.

Concept in a Minute:

Probability is the measure of the likelihood of an event occurring. It is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes.
Formula: Probability (Event) = (Number of favorable outcomes) / (Total number of possible outcomes)

Explanation:

Step 1: Identify the total number of pens in the lot.
The total number of pens is the sum of defective pens and good pens.
Total pens = Number of defective pens + Number of good pens
Total pens = 12 + 132 = 144 pens.

Step 2: Identify the number of favorable outcomes.
The question asks for the probability of picking a good pen. So, the favorable outcome is picking a good pen.
Number of good pens = 132.

Step 3: Calculate the probability.
Using the formula for probability:
Probability (Good pen) = (Number of good pens) / (Total number of pens)
Probability (Good pen) = 132 / 144.

Step 4: Simplify the fraction.
Both 132 and 144 are divisible by common factors. We can simplify this fraction to its lowest terms.
Divide both by 12:
132 ÷ 12 = 11
144 ÷ 12 = 12
So, the simplified probability is 11/12.

Final Answer: The probability that the pen taken out is a good one is 11/12.

Question:

It is given that in a group of 3 students, the probability of 2 students not having the same birthday is 0.992. What is the probability that the 2 students have the same birthday?

Concept in a Minute:

Complementary Events: Two events are complementary if they are mutually exclusive and their probabilities add up to 1. In simpler terms, if one event happens, the other cannot, and vice versa. The probability of an event not happening is 1 minus the probability of the event happening.

Explanation:

Let A be the event that 2 students do not have the same birthday.
Let B be the event that 2 students have the same birthday.

We are given the probability of event A, which is P(A) = 0.992.
The events A and B are complementary events because it is certain that either 2 students will not have the same birthday, or they will have the same birthday. There is no other possibility, and these two events cannot happen at the same time.

For complementary events, the sum of their probabilities is always 1.
So, P(A) + P(B) = 1.

We need to find the probability that the 2 students have the same birthday, which is P(B).

Using the property of complementary events:
P(B) = 1 – P(A)

Substitute the given value of P(A):
P(B) = 1 – 0.992

Calculate the result:
P(B) = 0.008

Therefore, the probability that the 2 students have the same birthday is 0.008.

Question:

A die is thrown once. Find the probability of getting a number lying between 2 and 6.

Concept in a Minute:

Explanation:

A standard die has six faces, numbered from 1 to 6. When a die is thrown once, the total number of possible outcomes is 6 (the numbers 1, 2, 3, 4, 5, 6).

We are interested in the probability of getting a number lying between 2 and 6. This means we are looking for numbers that are strictly greater than 2 and strictly less than 6. The numbers on the die that satisfy this condition are 3, 4, and 5.

Therefore, the number of favorable outcomes is 3.

Now, we can calculate the probability using the formula:
Probability = (Number of favorable outcomes) / (Total number of possible outcomes)
Probability = 3 / 6

This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 3.
Probability = 1 / 2

So, the probability of getting a number lying between 2 and 6 when a die is thrown once is 1/2.

Question:

A bag contains lemon flavoured candies only. Malini takes out one candy without looking into the bag. What is the probability that she takes out an orange flavoured candy?

Concept in a Minute:

Explanation:

The problem states that the bag contains lemon flavoured candies only. This means that there are no orange flavoured candies in the bag.
The event we are interested in is Malini taking out an orange flavoured candy.
The number of favorable outcomes (taking out an orange flavoured candy) is 0, because there are no orange flavoured candies in the bag.
The total number of possible outcomes is the total number of candies in the bag, which are all lemon flavoured. Let’s assume there are ‘N’ candies in the bag. So, the total number of possible outcomes is N.
Using the probability formula:
P(Orange flavoured candy) = (Number of orange flavoured candies) / (Total number of candies)
P(Orange flavoured candy) = 0 / N
P(Orange flavoured candy) = 0

Therefore, the probability that she takes out an orange flavoured candy is 0.

Question:

One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting a king of red colour.

Concept in a Minute:

Probability is calculated as the ratio of favorable outcomes to the total number of possible outcomes. In this case, we need to identify the number of kings with red color and the total number of cards in a standard deck.

Explanation:

A standard deck of 52 cards consists of four suits: hearts, diamonds, clubs, and spades. Each suit has 13 cards, including one king. The red suits are hearts and diamonds. Therefore, there are two kings of red color: the king of hearts and the king of diamonds.

The total number of possible outcomes is the total number of cards in the deck, which is 52.

The number of favorable outcomes (getting a king of red color) is 2.

The probability of an event is given by:
Probability = (Number of favorable outcomes) / (Total number of possible outcomes)

So, the probability of getting a king of red color is:
Probability (King of red color) = 2 / 52

This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 2.
Probability (King of red color) = 1 / 26

Question:

A die is thrown once. Find the probability of getting a prime number.

Concept in a Minute:

Probability is the measure of the likelihood that an event will occur. It is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes. For a fair die, the total possible outcomes are the numbers 1 through 6. Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves.

Explanation:

When a die is thrown once, the set of all possible outcomes is S = {1, 2, 3, 4, 5, 6}. The total number of possible outcomes is n(S) = 6.
We are interested in the event of getting a prime number. Let A be the event of getting a prime number.
The prime numbers in the set of possible outcomes are 2, 3, and 5.
So, the set of favorable outcomes for event A is A = {2, 3, 5}.
The number of favorable outcomes is n(A) = 3.
The probability of an event is given by the formula:
P(A) = (Number of favorable outcomes) / (Total number of possible outcomes)
P(A) = n(A) / n(S)
P(A) = 3 / 6
P(A) = 1/2

Therefore, the probability of getting a prime number when a die is thrown once is 1/2.

Question:

Five cards, the ten, jack, queen, king and ace of diamonds, are well-shuffled with their face downwards. One card is then picked up at random. What is the probability that the card is the queen?

Concept in a Minute:

Explanation:

The question asks for the probability of picking the queen of diamonds from a set of five specific diamond cards.

First, identify the total number of possible outcomes. The five cards are the ten, jack, queen, king, and ace of diamonds. Therefore, there are 5 possible outcomes when picking one card at random.

Next, identify the number of favorable outcomes. The favorable outcome is picking the queen of diamonds. There is only one queen of diamonds in the set. So, the number of favorable outcomes is 1.

Now, apply the formula for probability:
Probability (picking the queen) = (Number of favorable outcomes) / (Total number of possible outcomes)
Probability (picking the queen) = 1 / 5

Therefore, the probability that the card picked is the queen is 1/5.

Question:

One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting a spade.

Concept in a Minute:

Probability is defined as the ratio of favorable outcomes to the total number of possible outcomes. In a standard deck of 52 cards, there are four suits: spades, hearts, diamonds, and clubs. Each suit has 13 cards.

Explanation:

A standard deck of cards has 52 cards.
The deck is divided into four suits: spades, hearts, diamonds, and clubs.
Each suit has 13 cards.
The question asks for the probability of drawing a spade.
The number of favorable outcomes (drawing a spade) is the number of spades in the deck, which is 13.
The total number of possible outcomes is the total number of cards in the deck, which is 52.
The probability of an event is calculated as:
Probability = (Number of favorable outcomes) / (Total number of possible outcomes)
Therefore, the probability of getting a spade is:
Probability (Spade) = 13 / 52
This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 13.
13 ÷ 13 = 1
52 ÷ 13 = 4
So, the probability of getting a spade is 1/4.

Question:

Concept in a Minute:

Probability is calculated as the ratio of favorable outcomes to the total number of possible outcomes. Favorable outcome is the event we are interested in, and total outcomes are all the possible events.

Explanation:

First, determine the total number of coins in the piggy bank.
Number of 50 p coins = 100
Number of Rs 1 coins = 50
Number of Rs 2 coins = 20
Number of Rs 5 coins = 10

Total number of coins = 100 + 50 + 20 + 10 = 180

Next, identify the number of favorable outcomes, which is the number of 50 p coins.
Number of 50 p coins = 100

The probability of an event is given by the formula:
Probability = (Number of favorable outcomes) / (Total number of possible outcomes)

In this case, the probability of the coin being a 50 p coin is:
Probability (50 p coin) = (Number of 50 p coins) / (Total number of coins)
Probability (50 p coin) = 100 / 180

Simplify the fraction:
Probability (50 p coin) = 100 / 180 = 10 / 18 = 5 / 9

Therefore, the probability that the coin will be a 50 p coin is 5/9.

Question:

A box contains 5 red marbles, 8 white marbles, and 4 green marbles, One marble is taken out of the box at random. What is the probability that the marble taken out will be red?

Concept in a Minute:

Probability is the measure of the likelihood of an event occurring. It is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes.
Formula: Probability of an event (P(E)) = (Number of favorable outcomes) / (Total number of possible outcomes)

Explanation:

To find the probability of drawing a red marble, we first need to determine the total number of marbles in the box.
Number of red marbles = 5
Number of white marbles = 8
Number of green marbles = 4
Total number of marbles = Number of red marbles + Number of white marbles + Number of green marbles
Total number of marbles = 5 + 8 + 4 = 17

The favorable outcome in this case is drawing a red marble.
Number of favorable outcomes (drawing a red marble) = 5

Now, we can calculate the probability of drawing a red marble using the formula:
P(Red) = (Number of red marbles) / (Total number of marbles)
P(Red) = 5 / 17

Therefore, the probability that the marble taken out will be red is 5/17.

Question:

If P(E) = 0.05, what is the probability of ‘not E’?

Concept in a Minute:

The fundamental concept here is the relationship between the probability of an event and the probability of its complement. The probability of an event and the probability of its complement (the event not happening) always add up to 1. This can be expressed as P(E) + P(not E) = 1.

Explanation:

We are given the probability of an event E, denoted as P(E), which is 0.05.
We need to find the probability of ‘not E’, which is the probability that event E does not occur. This is often denoted as P(E’) or P(not E).
The rule of probability states that for any event E, the sum of the probability of the event occurring and the probability of the event not occurring is always equal to 1.
Mathematically, this is written as:
P(E) + P(not E) = 1
We are given P(E) = 0.05.
To find P(not E), we can rearrange the formula:
P(not E) = 1 – P(E)
Substitute the given value of P(E):
P(not E) = 1 – 0.05
P(not E) = 0.95
Therefore, the probability of ‘not E’ is 0.95.

Question:

A die is thrown once. Find the probability of getting an odd number.

Concept in a Minute:

Probability of an event is the ratio of the number of favorable outcomes to the total number of possible outcomes.
Probability = (Number of favorable outcomes) / (Total number of possible outcomes)

Explanation:

When a standard six-sided die is thrown once, the possible outcomes are the numbers on its faces.
The set of all possible outcomes (sample space) is S = {1, 2, 3, 4, 5, 6}.
The total number of possible outcomes is n(S) = 6.

We are interested in the event of getting an odd number.
The odd numbers in the sample space are {1, 3, 5}.
Let A be the event of getting an odd number.
The number of favorable outcomes for event A is n(A) = 3.

The probability of getting an odd number is given by the formula:
P(A) = n(A) / n(S)
P(A) = 3 / 6
P(A) = 1 / 2

Therefore, the probability of getting an odd number when a die is thrown once is 1/2.

Question:

A box contains 5 red marbels, 8 white marbles and 4 green marbles, One marble is taken out of the box at ramdom. What is the probability that the marble taken out will be white?

Concept in a Minute:

Explanation:

Step 1: Identify the total number of marbles in the box.
Number of red marbles = 5
Number of white marbles = 8
Number of green marbles = 4
Total number of marbles = 5 + 8 + 4 = 17

Step 2: Identify the number of favorable outcomes, which is the number of white marbles.
Number of white marbles = 8

Step 3: Calculate the probability of drawing a white marble.
Probability (white marble) = (Number of white marbles) / (Total number of marbles)
Probability (white marble) = 8 / 17

Therefore, the probability that the marble taken out will be white is 8/17.

Question:

Following experiment have equally likely outcomes? Explain.

A baby is born. It is a boy or a girl.

Concept in a Minute:

Equally likely outcomes refer to a situation where each possible outcome of an experiment has the same probability of occurring. In simpler terms, no outcome is more probable than any other.

Explanation:

This experiment of a baby being born having two possible outcomes: a boy or a girl. While biologically there might be very slight statistical differences in the probability of having a boy or a girl, for the purpose of a typical high school probability question, these two outcomes are considered to be equally likely. This is because the probability of having a boy is approximately 0.5, and the probability of having a girl is also approximately 0.5. Since these probabilities are equal, the outcomes are considered equally likely. Therefore, the experiment of a baby being born has equally likely outcomes.

Question:

Probability of an event E + Probability of the event ‘not E’ = _______.

Concept in a Minute:

The question is about the fundamental concept of complementary events in probability. Two events are complementary if one of them occurring means the other one cannot occur, and vice versa. The sum of probabilities of an event and its complement is always equal to 1.

Explanation:

Let E be an event. The event ‘not E’ is the complement of event E, often denoted as E’ or Eᶜ.
The definition of probability states that for any event A, the probability of A, denoted as P(A), is a number between 0 and 1, inclusive.
A fundamental property of probability is that the sum of the probabilities of an event and its complement is always 1. This is because an event either happens or it does not happen; there are no other possibilities.
Therefore, P(E) + P(not E) = 1.

The blank should be filled with the number 1.

Question:

The probability of an event that cannot happen is _________. Such as event is called _________.

Concept in a Minute:

Understanding probability: Probability is a measure of the likelihood of an event occurring, expressed as a number between 0 and 1. A probability of 0 means the event cannot happen, and a probability of 1 means the event is certain to happen.
Impossible events: An impossible event is an event that cannot occur under any circumstances.

Explanation:

The probability of an event that cannot happen is always 0. This is because there is no chance of this event occurring.
Such an event is called an impossible event.

Example: The probability of getting a 7 when rolling a standard six-sided die is 0. This is an impossible event because the numbers on a die are only 1, 2, 3, 4, 5, and 6.

Therefore, the probability of an event that cannot happen is 0. Such an event is called an impossible event.

Question:

The probability of an event is greater than or equal to ______ and less than or equal to ______.

Concept in a Minute:

The fundamental concept needed is the definition of probability. Probability quantifies the likelihood of an event occurring. It is always expressed as a value between 0 and 1, inclusive. A probability of 0 means the event is impossible, and a probability of 1 means the event is certain to occur.

Explanation:

The probability of any event, let’s call it P(E), must fall within a specific range. It cannot be less than 0, because you cannot have a negative chance of something happening. Similarly, it cannot be greater than 1, because you cannot have more than a 100% chance of something happening. Therefore, the probability of an event is always greater than or equal to 0 and less than or equal to 1.

The probability of an event is greater than or equal to 0 and less than or equal to 1.

Question:

One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting the queen of diamonds.

Concept in a Minute:

Probability is defined as the ratio of the number of favorable outcomes to the total number of possible outcomes. For a standard deck of 52 cards, each card has a unique identity.

Explanation:

A standard deck of 52 cards consists of 4 suits (hearts, diamonds, clubs, spades) and 13 ranks in each suit (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King).
We are interested in drawing the queen of diamonds.
There is only one queen of diamonds in a deck of 52 cards.
So, the number of favorable outcomes (drawing the queen of diamonds) is 1.
The total number of possible outcomes (drawing any card from the deck) is 52.
The probability of getting the queen of diamonds is calculated as:
Probability = (Number of favorable outcomes) / (Total number of possible outcomes)
Probability = 1 / 52.

Question:

Why is tossing a coin considered to be a fair way of deciding which team should get the ball at the beginning of a football game?

Concept in a Minute:

Probability, Fair Events, Equally Likely Outcomes

Explanation:

Tossing a coin is considered fair because it has two possible outcomes: heads or tails. Assuming the coin is unbiased, each outcome (heads or tails) has an equal probability of occurring, which is 1/2 or 50%. This means neither outcome is favored over the other. In a football game, one team is assigned heads and the other tails. Since each team has an equal chance of getting their assigned outcome, the decision is fair and unbiased, ensuring no team has an advantage from the start.

Next Chapter: Quadratic Equations

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